Area

Many students will already know how to calculate the area of squares
and rectangles. Some will also know how to compute the area of
parallelograms and triangles. The geoboard will help these students
strengthen their understanding of area even more. But rather than
give them formulas to calculate area, we will try to give them
conceptual tools that tend to persist in their minds long after the
formulas are gone.

For example, if a student were to take a pair of scissors and snip a
parallelogram along its height, she would quickly realize (by fitting
together the two pieces) that the area of a parallelogram is nothing
more than the area of the corresponding rectangle. Similarly, a right
triangle may be thought of as half a rectangle. These kinds of
associations help students understand area much better, and the
geoboard is an excellent device on which to illustrate these
concepts.

Still, the teacher must be able to calculate the area of certain
polygons on sight, since students will surely ask questions for which
you must know the answer (although it is sometimes wise not to give
it). We therefore list the common area formulas in Figure 5:

Figure 5. Common area formulas.

Note that each formula follows readily from geometric considerations:
the parallelogram and right triangle, for example, were mentioned
above. The formulas for arbitrary triangles are best understood by
cloning the given triangle and forming a parallelogram with the
resulting pair of congruent triangles. Finally, a trapezoid can be
divided into two triangles whose areas can be calculated separately
and added. Sample calculations are given in Figure 6:

Figure 6. Sample area calculations.

But what about an arbitrary polygon? For example, how do we find the
area of the eight-sided polygon depicted in Figure 7? After spending a
lot of time, perhaps days, calculating the area of squares, rectangles
and triangles, most students will break up a complicated polygon like
this one into more manageable pieces. In fact, take a moment to do
just that: break the eight-sided polygon in Figure 7 into squares and
triangles, and compute its area. Please, do it now - it's important
for what follows.